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Hilltop inflation

For $V(\phi)=V_0-m^2\phi^2/2$ with $V_0$ dominant,
$$
\eta_V\simeq-\frac{m^2M_{\rm Pl}^2}{V_0},
\qquad
\epsilon_V\simeq\frac12\eta_V^2\frac{\phi^2}{M_{\rm Pl}^2}.
$$
If $\widetilde N$ e-folds remain and $n_s-1\simeq2\eta_V$, then
$$
\phi=\phi_e e^{(n_s-1)\widetilde N/2},
\qquad
r\simeq2(n_s-1)^2\left(\frac{\phi_e}{M_{\rm Pl}}\right)^2e^{(n_s-1)\widetilde N}.
$$


Quadratic hilltop slow-roll prediction

Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 310 / 4 / d / Solution

Quadratic hilltop slow-roll prediction

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Quadratic hilltop slow-roll prediction

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